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12. Vorlesung WS 2006/07Softwarewerkzeuge1 V12 Stochastische Simulation zellulrer Signalprozesse Traditionelle numerische Anstze fr chemische/biochemische Kinetik: (a) beginne mit einem Satz von gekoppelten gewhnlichen Differentialgleichungen (fr die Reaktionsraten), die die zeitliche Entwicklung der Konzentrationen der beteiligten chemischen Stoffe angeben, (b) verwende einen geeigneten Integrationsalgorithmus um, basierend auf den Ratenkonstanten und einem Satz von Anfangsbedingungen, die Konzentrationen als Funktion der Zeit zu berechnen Erfolgreiche Anwendungen : fr den Hefe Zellzyklus, metabolic engineering, Modelle der gesamten metabolischen Pfade (E-cell),... Groes Problem: zellulre Prozesse laufen in sehr kleinen Volumina ab und es ist oft nur eine sehr kleine Anzahl an Moleklen beteiligt. z.B. Interagieren bei der Genexpression einige wenige Transkriptionsfaktor- molekle mit einer einzigen Gen-regulatorischen Region. E.coli Zellen enthalten nur etwa 10 Molekle des Lac Repressors.

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12. Vorlesung WS 2006/07Softwarewerkzeuge2 bercksichtige stochastische Effekte (Konsequenz1) die Modellierung der Reaktionen als kontinuierliche Flsse von Substanzen ist nicht mehr korrekt. (Konsequenz2) Es knnen erhebliche stochastische Fluktuationen auftreten. Fr die Untersuchung der stochastischen Effekte bei biochemischen Reaktionen werden stochastische Beschreibungen der chemischen Kinetik sowie Monte Carlo Computersimulationen benutzt. Daniel Gillespie (J Comput Phys 22, 403 (1976); J Chem Phys 81, 2340 (1977)) introduced the exact Dynamic Monte Carlo (DMC) method that connects the traditional chemical kinetics and stochastic approaches. Assuming that the system is well mixed, the rate constants appearing in these two methods are related.

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12. Vorlesung WS 2006/07Softwarewerkzeuge3 Dynamic Monte Carlo In the usual implementation of DMC for kinetic simulations, each reaction is considered as an event and each event has an associated probability of occurring. The probability P(E i ) that a certain chemical reaction E i takes place in a given time interval t is proportional to an effective rate constant k and to the number of chemical species that can take part in that event. E.g. the probability of the first-order reaction X Y + Z would be k 1 N x with N x :number of species X, and k 1 : rate constant of the reaction Similarly, the probability of the reverse second-order reaction Y + Z X would be k 2 N Y N Z. Resat et al., J.Phys.Chem. B 105, 11026 (2001)

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12. Vorlesung WS 2006/07Softwarewerkzeuge4 Dynamic Monte Carlo As the method is a probabilistic approach based on events, reactions included in the DMC simulations do not have to be solely chemical reactions. Any process that can be associated with a probability can be included as an event in the DMC simulations. E.g. a substrate attaching to a solid surface can initiate a series of chemical reactions. One can split the modelling into the physical events of substrate arrival, of attaching the substrate, followed by the chemical reaction steps. Resat et al., J.Phys.Chem. B 105, 11026 (2001)

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12. Vorlesung WS 2006/07Softwarewerkzeuge5 Basic outline of the direct method of Gillespie (Step i) generate a list of the components/species and define the initial distribution at time t = 0. (Step ii) generate a list of possible events E i (chemical reactions as well as physical processes). (Step iii) using the current component/species distribution, prepare a probability table P(E i ) of all the events that can take place. Compute the total probability P(E i ) : probability of event E i. (Step iv) Pick two random numbers r 1 and r 2 [0...1] to decide which event E will occur next and the amount of time by which E occurs later since the most recent event. Resat et al., J.Phys.Chem. B 105, 11026 (2001)

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12. Vorlesung WS 2006/07Softwarewerkzeuge6 Basic outline of the direct method of Gillespie Using the random number r 1 and the probability table, the event E is determined by finding the event that satisfies the relation Resat et al., J.Phys.Chem. B 105, 11026 (2001) The second random number r 2 is used to obtain the amount of time between the reactions As the total probability of the events changes in time, the time step between occurring steps varies. Steps (iii) and (iv) are repeated at each step of the simulation. The necessary number of runs depends on the inherent noise of the system and on the desired statistical accuracy.

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12. Vorlesung WS 2006/07Softwarewerkzeuge7 Bacterial Photosynthesis 101 Photons Light Harvesting Complexes light energy electronic excitation Reaction Center e H + pairs ATPase chemical energy cytochrome bc 1 complex H + gradient; transmembrane potential ubiquinon cytochrome c 2 electron carriers outside inside

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12. Vorlesung WS 2006/07Softwarewerkzeuge8 Modelling as metabolic network Chemical reactions involved:

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12. Vorlesung WS 2006/07Softwarewerkzeuge9 Photosynthesis cycle view light energy electronic excitation e H + pairs chemical energy H + gradient, transmembrane voltage outside inside The conversion chain: stoichiometries must match turnovers! electrons 2 cycles: protons

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12. Vorlesung WS 2006/07Softwarewerkzeuge10 LH1 / LH2 / RC im Lehrbuch Photonen einfangen Hu et al, 1998 B800, B850, Car. LH2: 8 Dimere LH1: 16 Dimere downhill Transport der Exzitonen LH2 LH1 RC

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